Theorem apsym 8699

Description: Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)

Assertion

Ref	Expression
apsym	|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

Proof of Theorem apsym

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8088	. . 3 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
2	1	adantl 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
3		cnre 8088	. . . . . 6 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
4	3	ad3antrrr 492	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
5		simplrl 535	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. RR )
6		simplrl 535	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> z e. RR )
7	6	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. RR )
8		reaplt 8681	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> ( x < z \/ z < x ) ) )
9	5, 7, 8	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> ( x < z \/ z < x ) ) )
10		reaplt 8681	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ x e. RR ) -> ( z # x <-> ( z < x \/ x < z ) ) )
11	7, 5, 10	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( z < x \/ x < z ) ) )
12		orcom 730	. . . . . . . . . . . 12 |- ( ( x < z \/ z < x ) <-> ( z < x \/ x < z ) )
13	11, 12	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( x < z \/ z < x ) ) )
14	9, 13	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> z # x ) )
15		simplrr 536	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. RR )
16		simplrr 536	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> w e. RR )
17	16	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. RR )
18		reaplt 8681	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> ( y < w \/ w < y ) ) )
19	15, 17, 18	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> ( y < w \/ w < y ) ) )
20		reaplt 8681	. . . . . . . . . . . . 13 |- ( ( w e. RR /\ y e. RR ) -> ( w # y <-> ( w < y \/ y < w ) ) )
21	17, 15, 20	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( w < y \/ y < w ) ) )
22		orcom 730	. . . . . . . . . . . 12 |- ( ( y < w \/ w < y ) <-> ( w < y \/ y < w ) )
23	21, 22	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( y < w \/ w < y ) ) )
24	19, 23	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> w # y ) )
25	14, 24	orbi12d 795	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> ( z # x \/ w # y ) ) )
26		apreim 8696	. . . . . . . . . 10 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
27	5, 15, 7, 17, 26	syl22anc 1251	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
28		apreim 8696	. . . . . . . . . 10 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
29	7, 17, 5, 15, 28	syl22anc 1251	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
30	25, 27, 29	3bitr4d 220	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
31		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> A = ( x + ( _i x. y ) ) )
32		simpllr 534	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> B = ( z + ( _i x. w ) ) )
33	31, 32	breq12d 4064	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
34	32, 31	breq12d 4064	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( B # A <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
35	30, 33, 34	3bitr4d 220	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> B # A ) )
36	35	ex 115	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
37	36	rexlimdvva 2632	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
38	4, 37	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> B # A ) )
39	38	ex 115	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
40	39	rexlimdvva 2632	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
41	2, 40	mpd 13	1 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2177   E.wrex 2486   class class class wbr 4051  (class class class)co 5957   CCcc 7943   RRcr 7944   _ici 7947   + caddc 7948   x. cmul 7950   < clt 8127   # cap 8674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675

This theorem is referenced by:  addext  8703  mulext  8707  ltapii  8728  ltapd  8731  aptap  8743  apdivmuld  8906  div2subap  8930  recgt0  8943  prodgt0  8945  irrmulap  9789  pwm1geoserap1  11894  absgtap  11896  geolim  11897  geolim2  11898  geo2sum2  11901  geoisum1c  11906  tanaddap  12125  egt2lt3  12166  sqrt2irraplemnn  12576  1sgm2ppw  15542  triap  16109  apdiff  16128